Fractional Higher Differentiability for Solutions of Stationary Stokes and Navier-Stokes Systems with Orlicz Growth

Abstract

We consider weak solutions (u,π):Ω→ℝn×ℝ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(u,\\pi ):{\\Omega }\ o \\mathbb {R}^{n}\ imes \\mathbb {R}$\\end{document} to stationary ϕ-Navier-Stokes systems of the type −diva(x,Eu)+∇π+[Du]u=fdivu=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$ \\left \\{ \\begin {array}{ll} -\\mathrm {div~} a(x,\\mathcal {E} u)+\ abla \\pi +[Du]u=f \\\\ \\mathrm {div~} u=0 \\end {array} \\right . $\\end{document} in Ω⊂ℝn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}${\\Omega }\\subset \\mathbb {R}^{n}$\\end{document}, and to the corresponding ϕ-Stokes systems, in which the convective term [Du]u does not appear. In the above system, the function a(x,ξ) depends Hölder continuously on x and satisfies growth conditions with respect to the second variable expressed through a Young function ϕ. The notation Eu\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal {E} u$\\end{document} is used for the symmetric part of the gradient Du. We prove results on the fractional higher differentiability of both the symmetric part of the gradient Eu\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal {E} u$\\end{document} and of the pressure π.